Image Processing Reference
In-Depth Information
pq
-
2
( -
pm
!
Rr
( ) =
(- 1)
m
pmq pmq
r
p
-2
m
(7.91)
pq
- 2
+
- 2
-
m
=0
m
!
!
!
2
2
These polynomials are orthogonal within the unit circle, so an analysed shape has to be re-
mapped to be of this size before calculation of its moments. The
orthogonality
of these
polynomials assures the
reduction
in the set of numbers used to describe a shape. More
simply, the radial polynomials can be expressed as
p
k
Rr
() =
Br
(7.92)
pq
pqk
kq
=
where the Zernike coefficients are
((
pk
+ )/2)!
pk
-
2
B
= (-1)
(7.93)
pqk
((
pk
- )/2)!(( + )/2)!(( - )/2)!
kq
kq
for
p
-
k
= even. The Zernike moments can actually be calculated from centralised moments
as
t
l
q
m
p
q
p
+ 1
t
Z
=
(-
j
)
m
B
µ
(7.94)
pq
pqk
( -2- + )( +2- )
k
l
q
m
q
l
m
kq
=
l
0=0
m
where
t
= (
k
-
q
)/2 and where
t
l
t
lt l
!
!( - )!
=
(7.95)
Analysis (and by using Equation 7.83, assuming
x
,
y
are constrained to the interval [-1, 1])
gives
=
µ
00
Z
00
=
2
(
Z
µ
-
j
µ
) = 0
(7.96)
11
01
10
=
3
(
Z
µ
-
j
2-
µ
µ
)
22
02
11
20
which can be extended further (Teague, 1980), and with remarkable similarity to the Hu
invariant moments (Equation 7.88).
The magnitude of these Zernike moments remains invariant to rotation which affects
only the phase; the Zernike moments can be made
scale
invariant by normalisation. An
additional advantage is that there is a reconstruction theorem. For
Nm
moments, the original
shape
f
can be reconstructed from its moments and the Zernike polynomials as
Nm
Σ Σ
fxy
(,
)
Z V
(,
xy
)
(7.97)
pq
pq
p
=0
q
These descriptors have been shown to good effect in application by reconstructing a good
approximation to a shape with only few descriptors (Boyce, 1983) and in recognition
(Khotanzad, 1990). There are
pseudo Zernike moments
(Teh, 1988) aimed to relieve the
